Problem 11 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 11-22, solve the quadratic equation by factoring. $$ x^{2}-2 x-8=0 $$

Step-by-Step Solution

Answer

The solution to the quadratic equation \(x^{2}-2 x-8=0\) by factoring are \(x = -2\) and \(x = 4\).

Step 1: Set up the equation

Firstly, set up the equation in the standard quadratic form \(ax^{2} + bx + c = 0\). Here, it is already given as \(x^{2} - 2x - 8 = 0\).

Step 2: Factor the quadratic equation

Factor the quadratic equation. Try to find two integers such that they add up to -2 (coefficient of x) and their product is -8. The integers -4 and +2 satisfy these conditions. So we can write \(x^{2} - 2x - 8 = 0\) as \((x + 2)(x - 4) = 0\).

Step 3: Set each factor equal to zero

To solve for x, each factor has to be set equal to zero. Hence, we get two equations: \(x + 2 = 0\) and \(x - 4 = 0\).

Step 4: Solve for x

Now, solve those two equations for x. This gives us \(x = -2\) from the equation \(x + 2 = 0\) and \(x = 4\) from the equation \(x - 4 = 0\).

Key Concepts

Factoring

Factoring is an essential technique used in algebra to simplify expressions and solve equations by expressing them as a product of their factors. In the context of quadratic equations, factoring involves identifying two binomial expressions whose multiplication yields the original quadratic expression. For the equation \(x^2 - 2x - 8 = 0\), our goal is to find two numbers that add up to -2 and multiply to -8.

Here's a straightforward way to think about it:

• The two numbers we're looking for are
  • "factors of \(-8\)"
  • which "sum to \(-2\)".

By testing pairs of factors of \(-8\), we discover that \( -4 \) and \( 2 \) work perfectly, since \( -4 + 2 = -2 \) and \( -4 \times 2 = -8 \). Thus, the quadratic can be factored as \((x + 2)(x - 4) = 0\).

Factoring not only simplifies the process of solving the equation but can also help in understanding the properties of the quadratic, such as identifying its roots.

Standard Form

The standard form of a quadratic equation is crucial for recognizing the basic structure and applying various algebraic techniques to solve it. A quadratic equation in its standard form is represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The given equation \(x^2 - 2x - 8 = 0\) is already written in this form.

In the standard form:

• \(a\) is the coefficient of \(x^2\).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

Understanding the standard form is important because it allows you to quickly identify the equation's coefficients, which are key to its factorization and the application of the quadratic formula, if needed. Furthermore, the standard form helps in determining the nature of the roots and the graph of the quadratic function.

Roots of Equations

Roots of an equation refer to the values of the variable that make the equation true, in this case, the values of \(x\) that satisfy \(x^2 - 2x - 8 = 0\). In simpler terms, the roots are points where the graph of the quadratic equation intersects the x-axis.

After factoring the quadratic as \((x + 2)(x - 4) = 0\), we set each factor equal to zero to find the roots. This gives us the equations:

• \(x + 2 = 0\)
• \(x - 4 = 0\)

Solving these equations, we find the roots:

• \(x = -2\) from \(x + 2 = 0\)
• \(x = 4\) from \(x - 4 = 0\)

These roots, \(x = -2\) and \(x = 4\), tell us where the parabola described by \(x^2 - 2x - 8 = 0\) crosses the x-axis. Understanding the concept of roots is fundamental for solving any quadratic equation and is a key concept in algebra and later calculus studies.